semblables

Semblables is the plural form of the French adjective semblable, meaning similar or alike. In ordinary French usage, semblable describes things that resemble each other. In a mathematical context, the term is used more precisely to denote figures that share the same shape, regardless of size.

In geometry, two figures are described as semblables when they are related by a similarity transformation. This consists of a combination of translation, rotation, reflection, and uniform scaling (dilation). As a result, corresponding angles are equal and corresponding sides are proportional. The constant of proportionality is the scale factor, the ratio of any corresponding side lengths.

Common criteria for similarity (especially with triangles) include:

- AA (angle–angle) criterion: if two corresponding angles are equal, the figures are semblables.

- SSS (side–side–side) criterion: if all three pairs of corresponding sides are proportional, the figures are semblables.

- SAS (side–angle–side) criterion: if two sides are proportional and the included angle is equal, the figures are semblables.

Key properties: semblables preserve shape but not size; they maintain angle measures and the ratios of corresponding sides. They are not necessarily congruent unless the scale factor is 1.

Aside from geometry, semblables can also appear in broader language to mean counterpoints or things that resemble one another, though this is more literary or philosophical than mathematical.

Etymology: semblable derives from Latin similis, via Old French semblable, with the plural form semblables in masculine and feminine use.